Question

Show that the expression pRT has the same units as the pressure, and thus that the ideal gas law is dimensionally consistent.

Answer 1

Answer and Explanation:

We have given the expression
`\rho RT`

Dimension of
`\rho =ML^(-3)`

Dimension of R which is gas constant
`=ML^2T^(-2)\Theta ^(-1)M^(-1)`

Dimension of temperature T
`\Theta ^(-1)`

And dimension of pressure
`ML^(-1)T^(-2)`

Now combine dimension of
`\rho RT`
`=ML^(-3)ML^2T^(-2)\Theta ^(-1)M^(-1)\Theta ^(-1)=ML^(-1)T^(-2)`

So the dimension of
`\rho RT` and dimension P is same so there unit will also be same

From ideal gas equation we know that
`PV=nRT`

`P=(n)/(V)RT=\rho RT`

As the both P and
`\rho RT` has same dimension so they are dimensionally constant